Philosophy from the mind of a fighter

Some Unfortunate Choices in Mathematics Terminology

Words can be tricky things. The same word can often carry wholly different meanings depending upon the context in which it is used. Take, for instance, the semantic range of the word “light.” This word can carry very different meanings when used in different contexts, as the following sentences illustrate.

That feather is light.

That shade of pink is light.

That laserbeam is light.

Each one of these sentences is of the form “That <noun phrase> is light,” but the word “light” intends an entirely different thing, in each. In (1), “light” is a description of the weight of the feather. In (2), “light” is a description of the intensity of the shade of pink. In (3), “light” is a description of the physical nature of the laserbeam. There is a well known fallacy of logic called equivocation which involves conflating such definitions in order to arrive at a false conclusion. For example, if I said…

Light things weigh less than heavy things

This shade of pink is light

Therefore, this shade of pink weighs less than heavy things

…my logic would be invalid, because the definitions of “light” used in (1) and (2) are completely different.

Mathematics, unfortunately, contains some terminology which tends to lead to these same sorts of equivocation fallacies, because the common usage of a word very often differs from the mathematical usage of that word. While there are numerous examples from which I could likely choose, today I’m going to focus on a case which I believe to be particularly egregious. Today, I’m going to discuss Real and Imaginary numbers.

If you continue your math education long enough, you will eventually encounter things which mathematicians refer to as Real numbers and Imaginary numbers. Because of these titles, people have come under the unfortunate and entirely incorrect misunderstanding that Real numbers are somehow better than Imaginary numbers, as if Reals correspond to the actual world more accurately than Imaginaries. Put succinctly, people think that Real numbers are real, and Imaginary numbers are just make-believe.

The bare truth of the matter is that Real numbers are no more real than Imaginary numbers are; and Imaginary numbers are no more make-believe than Real numbers are.

This is why there are a significant number of mathematicians (including me) who chafe at the phrase “Imaginary numbers.” We prefer to use the synonymous phrase “Complex numbers.” Personally, even if everyone were to switch to saying “Complex numbers,” I feel that “Real number” is still a rather misleading phrase, but unfortunately that particular bit of terminology is so deeply rooted in mathematical culture that it’d be almost impossible to dig it out and replace it.

It is usually at this point in the discussion when one of my friends says to me, “but Imaginary numbers rely on , and that is not a real number!” This is where the confusion tends to set in, because I always answer them by saying, “Actually, is a real number, it’s just not a Real number.” My response is generally met with exasperation and quizzical stares. Unfortunately, that capital-R doesn’t really stand out, in spoken language.

In order to better explain what I mean, let me jump backwards a couple thousand years. There used to be a grand mystery, among mathematicians, regarding the diagonal of a square. Basically, mathematicians wanted to find the proportion of a square’s diagonal to one of its sides. They were convinced that there was an actual number which would satisfy this proportion, but no one had yet been able to discover it. They knew that squaring this proportion (in a unit square) would yield a value of 2, thanks to the Pythagorean Theorem, but no one had been able to calculate the precise number that could satisfy the proposition. Eventually, these mathematicians came to a shocking realization– I have written before about one of the legends which rose up in the wake of this discovery. Regardless of whether these legends actually occurred, it is true that these mathematicians eventually had to deal with an incredible fact: there existed no number which could represent the proportion of a square’s diagonal to one of its sides. To the ancient Greeks, was not a number.

So, mathematicians redefined their concept of “number.” They began to incorporate concepts which could not be represented as a ratio of whole numbers, like , calling them “irrational.” Originally, the word “irrational” simply meant “unable to be expressed as a ratio,” but the extremely counter-intuitive nature of such numbers is the very reason that the word “irrational” has come to be synonymous with “lacking in reason.” Some philosophers and mathematicians stood in very vocal opposition to the idea that a number could be “irrational.” It simply didn’t make any sense to them. However, as time wore on, and the field of mathematics started to become irrevocably entwined with the concept of irrational numbers, the naysayers began to fall away, and eventually it became nigh universally accepted that irrational numbers were every bit as real and legitimate as rational numbers.

However, it would not take long before another curious puzzle emerged. Mathematicians knew that squaring any number, whether positive or negative (and negative numbers had their own fight for acceptance, by the bye), would yield a positive result. So, then, what would happen if we try to find the square root of a negative number? Most mathematicians thought this was a completely preposterous notion, and asserted that it simply could not be done. They proclaimed that, if some formula required that you take the square root of a negative number, you must have done something wrong. However, some mathematicians began to play an interesting game. If we assume that there exists some number such that , then we are suddenly able to resolve any equation which requires us to take the square root of a negative number. This quickly became an incredibly useful and widely utilized tool.

However, just as before, philosophers and mathematicians began to cry out against this usage. They considered such a notion to be a terrible affront to reason, a horrific departure from true understanding; but the power of this new tool was completely undeniable. As a result, some mathematicians really wanted to use it, despite their philosophical aversion to it. One such person was the eminent René Descartes, and it is him we have to blame for the “imaginary” appellation. Descartes advocated the use of this , but argued that it should only be considered an imaginary thing, a conceptual tool which did not actually reflect reality. This seemed like a good enough justification to most mathematicians, and so the use of became widespread and popular. Some mathematicians argued it was a true number, others were content to simply call it “imaginary,” but it was not long before all recognized its utility. All the numbers they used which could be represented without became known as the Real numbers, while the rest became known as Imaginary numbers.

As it turns out, actually has some very important applications to the real world. This number actually describes real world quantities just like natural numbers, negative numbers, rational numbers, irrational numbers, and transcendental numbers do. Take, as an example, the Wave Function, which is utilized in physics: . The in that formula is , and it is absolutely necessary to particle physics. It represents an actual quantity describing an actual property of a physical thing which actually exists. This is no less real than any Real number.

The unfortunate terminology has led to a lot of confusion. Even today, it’s still common to hear mathematics and physics teachers tell their students that “Imaginary numbers” are just make-believe things while the “Real numbers” are truly real. The truth of the matter is that all numbers are things which humanity has invented in order to attempt to describe the patterns which are found in the world around us. The number 2 is not any more real than the number or the number 0 or the number -2 or the number or the number .

The current school math explanation:
You take the a and distribute it to the b to get ab
and then you distribute the a to the c to get ac
and then you add them together to get ab + ac

I have come across this explanation in several places, and once again real damage is done to the language, and real confusion sown. “Distribute” means “to spread or share out” as in “The Arts Council distributed its funds unevenly, as usual. Opera got the lion’s share.” So it is NOT the a that is distributed. I tried to find a definition of the term in wordy form as it applies to algebra systems but failed. Heavy thinking produced the “answer”. What is being distributed is the second factor on the left.
Example:
Take 3 x 7. We know that the value of this is 21
Distribute, or spread out, the 7 as 2 + 5 . . . . . . . . the b + c
Then 3 x (2 + 5) has the value 21
But so does 3 x 2 + 3 x 5. To check, get out the blocks !
So 3 x (2 + 5) = 3 x 2 + 3 x 5 ……… The Law !

Regarding the “second” version of the distributive property, a(b – c) = ab – ac, this cannot just be stated, and you won’t find it in any abstract algebra texts. Since the students are looking at this before they have encountered the signed number system, a proof must not involve negative numbers, as a, b and c are all natural numbers. It can be done, and it will be put on my next “Language in math” post.

I have tried in introducing the different kinds of numbers to say “complex-valued numbers” or “real-valued numbers”, which I like to think emphasizes that these are all numbers and there’s just different kinds of values they have. I’m not sure this changes things much, though, since “complex” particularly does carry a connotation of “too hard to work with except when there’s no way around it”. But, heck, even something as familiar as “negative numbers” has a, well, negative overtone to it.

One of the problems of “accepting the meaning” results from the popular idea of extensions of the number system. The only extension is the incorporation of zero into the natural numbers. When fractions are introduced we are creating a new number system, for measurement of quantity, and all the numbers are “fractions”, with 3/1 being the 3 in the new system, and showing the mapping of the natural numbers into the fractional numbers. More trouble is caused by considering the negative numbers as an extension of the (now we have to call them positive) numbers. So bad! In fact the new system can and is (sometimes) called the signed numbers, created for measurements of position and change, and not meaningful as measures of quantity.
The feature of the “complex !!!” number system which is often overlooked is that every point in a plane can be described (measured) with a complex number, and this is where it stops. In 3D we have vectors, but lose one of the features of numbers, the commutative law of multiplication.

If the positive integers are seen as the prototypical examples of numbers, then the complex numbers are further from the prototype than the real numbers in quite a significant way, since the field of the complex numbers cannot be ordered, but the real numbers can be. This could be part of the reason people resist thinking of the complex numbers as proper numbers.

This made me think of complex analysis and elementary analysis. Complex analysis doesn’t mean that it is “complex”, though it can become very difficult, it just means that complex numbers are used in the analysis. On the other hand, elementary analysis doesn’t mean that it is simple since it just means that it has no complex numbers in it 🙂 It’s quite confusing for people who don’t know about them.

Using the word Imaginary to describe Complex numbers went out of style in the mid 19th Century until rehashed a few years ago by a Steve, who used the word to describe events that he says defy all the laws of physics. His untestable theory that defies physics is taught as fact in most western schools. Another guy making a bunch of money using a very Pythagorean Platform.